Date/Time : 25 Jan 2022 16:00
Venue : via Zoom
Speaker : Feng Zhu, Technion - Israel Institute of Technology

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Meeting ID: 875 1147 6637
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Abstract: Putting a hyperbolic metric on a closed surface gives us a linear representation (the holonomy representation) with many nice geometric and dynamical properties. Anosov representations generalise this picture, or more generally the theory of convex cocompact subgroups of rank-one Lie groups such as PSL(2,R) or PSL(2,C), to higher-rank Lie groups such as PSL(d,K) for d>2. They give us an open subset of the space of representations consisting of quasi-isometric embeddings of word-hyperbolic groups.

There are many equivalent characterisations of Anosov representations; in particular, Labourie’s original definition uses a flow coming from the hyperbolic group to obtain additional control over the image subgroup. In this talk, I will introduce a relative notion of Anosov representations, similarly formulated in terms of flows, which allows for (certain) parabolic elements / controlled breakdowns of hyperbolicity.

This generalises work of Canary–Zhang–Zimmer and is joint work in progress with Andrew Zimmer.

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Date/Time : 18 Jan 2022 10:00
Venue : via Zoom
Speaker : Ara Basmajian, City University of New York

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Meeting ID: 875 1147 6637
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Abstract:  There are various contexts for which it is natural to study spaces that look the same from any point of the space (that is, “homogeneous spaces”). For example, the simply connected Riemann surfaces, namely the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact they are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being K-quasiconformal (has eccentricity bounded by K). Conformal homeomorphisms are 1-quasiconformal.

A Riemann Surface X is said to be K-quasiconformally homogeneous if for any two points x and y on it, there exists a K-quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH Riemann surface. After introducing the basics, the focus of this talk will be on Riemann surface structures that are QCH, and their connections to the topology and hyperbolic geometry of the surface.

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Date/Time : 12 Jan 2022 16:00
Venue : via Zoom
Speaker : Jia Jia, National University of Singapore

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This is a joint work with Doctor Takahiro Shibata and Professor De-Qi Zhang. The potential density problem of varieties over number fields asks whether the set of rational points of becomes Zariski dense after a finite extension of the base field. In this talk, I will explain how to get potential density from some dynamic conditions on the varieties. I will also talk about the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.

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Date/Time : 10 Nov 2021 16:00
Venue : via Zoom
Speaker : Maxime Wolff, Sorbonne University

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Meeting ID: 875 1147 6637
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Abstract: In this talk I will present joint works with Kathryn Mann, in which we take advantage of strong rigidity properties of the actions of some Fuchsian groups on the circle. From this, we derive rigidity properties of some larger groups acting on the circle, such as mapping class groups of pointed surfaces, as well as the existence of groups of critical regularity.

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Date/Time : 03 Nov 2021 16:00
Venue : via Zoom
Speaker : Zhong Guolei, National University of Singapore

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Meeting ID: 875 1147 6637
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Abstract:  We consider a projective klt pair (X,D) of arbitrary dimension with the anti-log canonical divisor -(K+D) being strictly nef. In this talk, I will first show that such X is rationally connected. In particular, when X is a threefold, we show that -(K+D) is indeed ample, which gives a positive answer to the singular version of Campana-Peternell’s conjecture (for the threefold case). Finally, we extend our first result to the (relative) fibration case and study the geometric structure of this fibration. This is a joint work with Jie Liu, Wenhao Ou, Juanyong Wang and Xiaokui Yang.

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Date/Time : 27 Oct 2021 16:00
Venue : via Zoom
Speaker : Beatrice Pozzetti, University of Heidelberg

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Meeting ID: 875 1147 6637
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Abstract:  Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of $\Theta$ positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of $\Theta$ positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.

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Date/Time : 13 Oct 2021 16:00
Venue : via Zoom
Speaker : Jonas Beyrer, University of Bonn

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Meeting ID: 875 1147 6637
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Abstract:  The study of higher rank Teichmüller spaces has attracted a lot of attention in the last 20 years. Those spaces are connected components of the character variety Hom(H,G)/G that consist entirely of discrete and faithful representations; here G is a Lie group and H the fundamental group of a closed surface. Those components generalize classical Teichmüller space, the case G=PSL(2,R), which also identifies with the moduli space of hyperbolic structures on the surface. Representations in higher rank Teichmüller spaces, so called „positive representations“, are not only discrete and faithful, but have many other striking properties and similarities to hyperbolizations (e.g. constant entropy, collar lemmas, etc). However surprisingly there are many representations that are not in higher rank Teichmüller spaces and still share many of this striking properties: representations that satisfy a „partial positivity“. In this talk on joint work with Beatrice Pozzetti I want to discuss partially positive representations and their main difference to representations in higher rank Teichmüller spaces: degenerations.

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Date/Time : 06 Oct 2021 16:00
Venue : via Zoom
Speaker : Purvi Gupta, Indian Institute of Science, Bangalore

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Meeting ID: 875 1147 6637
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Abstract: A compact subset of $\mathbb{C^n}$ is said to be polynomially convex if it is defined by a family of polynomial inequalities. Polynomial convexity grants certain approximation-theoretic properties to the underlying set. When the set is a real submanifold of $\mathbb{C^n}$, its convexity properties are partly influenced by its topology, and the local and global structure of its CR (complex-real) singularities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although there are some obvious bounds). In this talk, we will discuss some recent improvements on the previously known bounds, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.

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Date/Time : 30 Sep 2021 10:00
Venue : via Zoom
Speaker : Brian Collier, University of California Riverside

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Meeting ID: 875 1147 6637
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Abstract:  In this talk, we will define a special class of nilpotent elements of complex semisimple Lie algebra and describe how a Slodowy slice construction defines connected components of moduli spaces of G-Higgs bundles on a compact Riemann surface for specific real groups G. Through the nonabelian Hodge correspondence, these components define connected components of the G-character variety with many nice features. We will then describe how these nice features, together with recent work of Guichard-Labourie-Wienhard on positive representations imply that such components are higher Teichmuller spaces, that is, they consist entirely of discrete and faithful representations. In the remaining time, we will describe what remains to be proven to finish the classification of higher Teichmuller spaces.

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